Question

Approximately 14 percent of the population of Arizona is 65 years or older. A random sample of five persons from this population is taken. The probability that less than 2 of the 5 are 65 years or older is :

A)0.8533

B) 0.1467

C) 0.4704

D) 0.3829

E) None of the above.

Answer 1

A)0.8533

Explanation:

For each person, there are only two possible outcomes. Either they are 65 or older, or they are not. The probability of a person being 65 or older is independent of any other person. So we use the binomial probability distribution to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

In which
`C_(n,x)` is the number of different combinations of x objects from a set of n elements, given by the following formula.

`C_(n,x) = (n!)/(x!(n-x)!)`

And p is the probability of X happening.

14 percent of the population of Arizona is 65 years or older.

This means that
`p = 0.14`

A random sample of five persons from this population is taken.

This means that
`n = 5`

The probability that less than 2 of the 5 are 65 years or older is :

`P(X < 2) = P(X = 0) + P(X = 1)`

In which

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

`P(X = 0) = C_(5,0).(0.14)^(0).(0.86)^(5) = 0.4704`

`P(X = 1) = C_(5,1).(0.14)^(1).(0.86)^(4) = 0.3829`

`P(X < 2) = P(X = 0) + P(X = 1) = 0.4704 + 0.3829 = 0.8533`

So the correct answer is:

A)0.8533